36 problems

Let $\pi: M \rightarrow S$ be a smooth fibration, with total space $M$, and let $F\to M$ be a flat complex vector bundle. The Wagoner conjecture. Reidemeister–Franz torsion and Ray…

Let $M$ be a closed manifold and let $M\to E\xrightarrow{\pi}B$ be an $M$-bundle over a closed $3$-manifold $B$. A semi-$s$-cobordism from $B$ to $B'$ is a bordism $(V,B,B')$ with…

Let $M\to E\to B$ be a smooth fiber bundle whose fiber admits a negatively curved Riemannian metric. Assume that $B$ is a paracompact, Hausdorff space with the homotopy type of a s…

Semi-bundle semi-transition conjecture. On pairwise overlaps,

Fukaya–Yamaguchi's fibering conjecture. A finite cover of $M$ fibers over a nilmanifold with a fiber that has nonnegative Ricci curvature and finite fundamental group.

Homological triviality conjecture. The conclusion of the proposition for trivial symplectic bundles should hold for every Hamiltonian bundle: every…

Let $Y$ be a complex analytic manifold and let $f:Y\to\Delta$ be a projective morphism, where $\Delta$ is a disc. A morphism is a homotopy fiber bundle if it has the homotopy-local…

Let $Sigma$ be a compact complex manifold of dimension greater than $n$, with $ngeq 2$, and let $\mathcal{F}$ be the horizontal foliation. Let $\mathcal{N}$ be the bundle equipped…

Let $M$ be a compact Kähler manifold of dimension $k$, and let $E$ be a holomorphic $\mathbb{B}^n$-bundle over $M$ embedded in the associated $\mathbb{CP}^n$-bundle $\widehat{E}$.…

Associated-connection conjecture. The connection $\eta$ can be seen as an associated connection to a principal connection on $(\hat{P}, M, \pi_{\hat{P}, M}, \operatorname{Aut}(G))$…

Let $F$ be a smooth manifold homeomorphic to … Such a manifold satisfies Property (S) when it has the spinnability property for fiber bundles defined earlier in the paper. Property…

Richmond–Slofstra's conjecture. The element $w$ has a Grassmannian BP decomposition.

Let $N\to B$ be a fiber bundle with base $B$ a connected closed manifold, and with all fibers diffeomorphic to a fixed connected closed manifold $Z$. Suppose there is no fiberwise…

Fiber-bundle conjecture. If all points in $S$ are weakly $m$-strained, then $\phi:X\to S$ is a fiber bundle map.

Let $p\colon E_{\Sigma_F}\to B$ be a toric variety bundle, where $E\to B$ is a $T_F$-principal bundle over a smooth complete base $B$ and $\Sigma_F$ is a smooth complete fan. For e…

Fiber-bundle conjecture. An M-ideal of any rank in $\mathscr{P}(\mathscr{A})$ will give rise to a fiber bundle whose fiber is homeomorphic to the complement of an affine abelian ar…

Let $S$ be the closed surface underlying the quasi-Hitchin representations under consideration, let $M_{\mathbb C}$ be the manifold associated with a quasi-Hitchin representation i…

Torus-fibration conjecture. $M$ is finitely covered by a manifold diffeomorphic to a fiber bundle over an $m$-dimensional torus.

Let $S^G$ be a Yang-Mills-type action functional with gauge group $G$, let $\hat{G}$ be an extension of $G$, and let $\operatorname{Ext}(S^G;\hat{G})$ denote the continuous…

Let $(V,\partial V)$ be a manifold of dimension at least $3$. Let $X\to B\to V$ be a bundle with structure group $\operatorname{Homeo}(X)$, where $X$ is any manifold with a point-s…

K-theory presentation conjecture. The ring $K^*(E(X))$ is a free $K^*(B)$-module of rank $m=\chi(X)$ and is isomorphic to $\mathfrak{S}/\mathfrak{J}$.

Geometric Cheeger conjecture. If a Cheeger-type inequality for SynCut exists, then it will reflect the global geometric information encoded by geometric quantities associated with…

An invariant connection on a $G$-structure is understood in the setting of fat bundles over compact homogeneous spaces. Invariant-connection extension conjecture. The classificatio…

Let $X$ be a simply connected closed manifold or a simply connected finite simplicial complex, and let $p\colon E\to X$ be a fiber bundle with closed-manifold fiber $M$. Assume tha…

Let $X$ be a closed manifold or finite simplicial complex, and let $p\colon E\to X$ be a fiber homotopically trivial bundle with closed-manifold fiber $M$. Assume that $E$ admits a…